Horner's Method - Description of The Algorithm

Description of The Algorithm

Given the polynomial

where are real numbers, we wish to evaluate the polynomial at a specific value of, say .

To accomplish this, we define a new sequence of constants as follows:

$\begin{align} b_n & := a_n \\ b_{n-1} & := a_{n-1} + b_n x_0 \\ & {}\ \ \vdots \\ b_0 & := a_0 + b_1 x_0. \end{align}$

Then is the value of .

To see why this works, note that the polynomial can be written in the form

Thus, by iteratively substituting the into the expression,

$\begin{align} p(x_0) & = a_0 + x_0(a_1 + x_0(a_2 + \cdots + x_0(a_{n-1} + b_n x_0)\cdots)) \\ & = a_0 + x_0(a_1 + x_0(a_2 + \cdots + x_0(b_{n-1})\cdots)) \\ & {} \ \ \vdots \\ & = a_0 + x_0(b_1) \\ & = b_0. \end{align}$

Read more about this topic: Horner's Method

Famous quotes containing the words description of the, description of and/or description:

“The next Augustan age will dawn on the other side of the Atlantic. There will, perhaps, be a Thucydides at Boston, a Xenophon at New York, and, in time, a Virgil at Mexico, and a Newton at Peru. At last, some curious traveller from Lima will visit England and give a description of the ruins of St. Paul’s, like the editions of Balbec and Palmyra.”
—Horace Walpole (1717–1797)

“It is possible—indeed possible even according to the old conception of logic—to give in advance a description of all ‘true’ logical propositions. Hence there can never be surprises in logic.”
—Ludwig Wittgenstein (1889–1951)

“I fancy it must be the quantity of animal food eaten by the English which renders their character insusceptible of civilisation. I suspect it is in their kitchens and not in their churches that their reformation must be worked, and that Missionaries of that description from [France] would avail more than those who should endeavor to tame them by precepts of religion or philosophy.”
—Thomas Jefferson (1743–1826)